In view of the recent discussions about flaws in arguments and logical
fallacies, I have attached an introduction to logic. I found this
somewhere on the net a while ago, modified it and updated it a bit. I
think anyone involved in Biblical studies should be familiar with basic
logic so that they can present good sound arguments instead of coloured
interpretations and pure speculations.

There is a lot of very poor argument in modern Biblical studies (and
other fields as well no doubt) even from very well known and high
profile scholars. This document attempts to provide a basic introduction
to logic in order to guide readers in constructing a careful logical
argument that can stand up to scrutiny as well as helping readers to
spot poor and invalid arguments.

Logic is the science of reasoning, proof, thinking, or inference
[Concise OED]. Logic will let you analyze an argument or a piece of
reasoning, and work out whether it is correct or not. To use the
technical terms, logic lets you work out whether the reasoning is
valid or invalid.

Note also that this document deals only with simple boolean logic.
Other sorts of mathematical logic, such as fuzzy logic, obey different
rules. When people talk about logical arguments, though, they usually
mean the type being described here.

One problem with boolean logic is that people don't have to be
consistent in their goals and desires. People use fuzzy logic and
non-logical reasoning to handle their conflicting goals; boolean logic
isn't good enough. For example:

"John wishes to speak to the person in charge. The person in charge
is Steve. Therefore John wishes to speak to Steve."

Logically, that's a totally valid argument. However, John may have a
conflicting goal of avoiding Steve, meaning that the answer obtained
by logical reasoning may be inapplicable to real life. Garlic tastes
good, strawberry ice cream tastes good, but strawberry garlic ice
cream is only logically a good idea.

Sometimes, principles of valid reasoning which were thought to be
universal have turned out to be false. For example, for a long time
the principles of Euclidean geometry were thought to be universal
laws.

However, keeping those caveats and limitations in mind, let's go on to
consider the basics of boolean logic.

2. Basic concepts

The building blocks of a logical argument are propositions, also
called statements. A proposition is a statement which is either true
or false. For example:

"The first Holden car was built in 1948."

"Ginger cats are always male."

"Canberra is the capital of Australia."

Propositions may be either asserted (said to be true) or denied (said
to be false).

Note: This is a technical meaning of the word "deny", not the everyday
meaning.

When a proposition has been asserted based on some argument, we
usually say that it has been affirmed.

The proposition is generally viewed as the meaning of the statement,
and not the particular arrangement of words used. So "An even prime
number greater than two exists" and "There exists an even prime number
greater than two" both express the same (false) proposition.

Sometimes, however, it is better to consider the wording of the
proposition as significant, and use linguistic rules to derive
equivalent statements if necessary.

3. What is an argument?

There are three stages to an argument: premises, inference, and conclusion.

Stage 1: Premises

For the argument to get anywhere, you need one or more initial
propositions. These initial statements are called the premises of the
argument, and must be stated explicitly.

You can think of the premises as the reasons for accepting the
argument, or the evidence it's built on. Premises are often indicated
by phrases such as "because", "since", "let's assume",
and so on.

Stage 2: Inference

Next the argument continues step by step, in a process called
inference.

In inference, you start with one or more propositions which have been
accepted. You then use those propositions to arrive at a new
proposition. The new proposition can, of course, be used in later
stages of inference.

There are various kinds of valid inference -- and also some invalid
kinds, but we'll get to those later. Inference is often denoted by
phrases such as "implies that" or "therefore".

Stage three: Conclusion

Finally, you arrive at the conclusion of the argument, another
proposition. The conclusion is often stated as the final stage of
inference.

The conclusion is affirmed on the basis the original premises, and the
inference from them. Conclusions are often indicated by phrases such
as "therefore", "it follows that", "we conclude" and so on.

***********
(The phrase "obviously" is often viewed with suspicion, as it gets
used to intimidate people into accepting things which aren't true at
all. If something doesn't seem obvious to you, don't be afraid to
question it. You can always say "Oh, yes, you're right, it is obvious"
when you've heard the explanation.)
***********

4. Types of argument

There are two traditional types of logical argument: deductive and
inductive.

1. A deductive argument is one which provides conclusive proof of its
conclusions. It is either valid or invalid.
A valid deductive argument is defined as one where if the premises
are true, then the conclusion must also be true.
2. An inductive argument is one where the premises provide some
evidence for the truth of the conclusion.
Inductive arguments are not valid or invalid, but we can talk
about whether they are better or worse than other arguments. We
can also discuss how likely their premises are.

There are forms of argument in ordinary language which are neither
deductive nor inductive. However, we'll concentrate on deductive
arguments, as they are often viewed as the most rigorous and
convincing.

Here is an example of a deductive argument:

* Premise: Every event has a cause
* Premise: The universe has a beginning
* Premise: All beginnings involve an event
* Inference: This implies that the beginning of the universe
involved an event
* Inference: Therefore the beginning of the universe had a cause
* Conclusion: The universe had a cause

Note that the conclusion of one argument might be a premise in another
argument. A proposition can only be a premise or a conclusion of a
particular argument; the terms don't make sense in isolation.

5. Recognizing an argument

Sometimes arguments won't follow the order described above. For
instance, the conclusions might be stated first, and the premises
stated afterwards in support of the conclusion. This is perfectly
valid, if sometimes a little confusing.

Arguments are harder to recognize than premises or conclusions. Lots
of people shower their writing with assertions, without ever producing
anything you might reasonably call an argument.

To make the situation worse, some statements look like arguments but
are not. For example:

"If the Bible is accurate, Jesus must either have been insane, an
evil liar, or the Son of God."

The statement above isn't an argument; it's a conditional statement.
It doesn't assert the premises which are needed to support what looks
like its conclusion. (Even if you add those assertions, it still
suffers from a number of other logical flaws.)

Here's another example:

"God created you; therefore obey and worship God."

The phrase "obey and worship God" is neither true nor false. Therefore
it isn't a proposition, and the sentence isn't an argument.

Causality is important as well. Suppose we're trying to argue that
there's something wrong with the engine of a car. Let's look at two
statements of the form "A because B". Here's the first:

"The car won't start because there's something wrong with the
engine."

That's not an argument for there being something wrong with the
engine; it's an explanation of why the car won't start. We're
explaining A, using B as the explanation.

Now consider a second statement:

"There must be something wrong with the engine of the car, because
it won't start."

Here we're arguing for A, giving B as evidence. The statement "A
because B" is an argument.

The difference between the two cases might not be completely clear.
So, remember that "A because B" is equivalent to "B therefore A". The
two statements then become:

We're supposed to be arguing that there's something wrong with the
engine, but now it's obvious that the first statement doesn't do that at
all. Only the second statement is arguing that there's something wrong
with the engine.

6. Implication

There's one very important thing to remember:

The fact that a deductive argument is valid doesn't necessarily
mean that its conclusion holds.

That may seem confusing, but it's because of the slightly
counter-intuitive nature of how implication works.

Obviously you can build a valid argument out of true propositions. But
you can also build a completely valid argument using only false
propositions. For example:

* If the premises are false and the inference valid, the conclusion
can be true or false. (Lines 1 and 2.)
* If the premises are true and the conclusion false, the inference
must be invalid. (Line 3.)
* If the premises are true and the inference valid, the conclusion
must be true. (Line 4.)

A sound argument is a valid argument whose premises are true. A sound
argument therefore arrives at a true conclusion. Be careful not to
confuse sound arguments with valid arguments.

Ultimately, the conclusion of a valid logical argument is only as
compelling as the basic premises it is derived from. Logic in itself
does not solve the problem of verifying the basic assertions which
support arguments. The only way to verify basic assertions is by
scientific enquiry.

7. Fallacies

In everyday English the word "fallacy" is used to refer to mistaken
beliefs, as well as to the faulty reasoning that leads to those
beliefs. In logic, the term is generally used for a form of
technically incorrect argument -- especially if the argument appears
valid or convincing.

So for the purposes of this discussion, a fallacy is a logical
argument which looks correct, but which can be seen to be incorrect
when examined more carefully. If fallacies are recognized they can
be pointed out as being fallacious and will therefore be less likely
to mislead people.

Below is a list of some common fallacies, and also some rhetorical
devices often used in debate. The list isn't intended to be
exhaustive.

Sadly, many of the examples below have been taken directly from
Usenet, though some have been rephrased for the sake of clarity.

One of the simplest fallacies is to rely on anecdotal evidence. For
example:

"Violent crime is on the increase because you hear a lot more about
it on the news these days."

It's quite valid to use personal experience to illustrate a point; but
such anecdotes don't really prove anything to anyone. Your friend may
say he met Elvis in the supermarket, but those who haven't had the
same experience will require more than your friend's anecdotal
evidence to convince them.

Argumentum ad baculum / Appeal to force

An Appeal to Force happens when someone resorts to force (or the
threat of force) to try and push others to accept a conclusion. This
fallacy is often used by politicians, and can be summarized as "might
makes right". The threat doesn't have to come directly from the person
arguing. For example:

"If you don't turn to Jesus Christ, you'll burn in Hell!"

"... In any case, I know your phone number and I know where you
live. Have I mentioned I am licensed to carry concealed weapons?"

Argumentum ad hominem

Argumentum ad Hominem literally means "argument directed at the man".
There are two types, abusive and circumstantial.

If you argue against some assertion by attacking the person who made
the assertion, then you have committed the abusive form of argumentum
ad hominem. A personal attack isn't a valid argument, because the
truth of an assertion doesn't depend on the virtues of the person
asserting it. For example:

"No intelligent person could believe in Creation."

Sometimes in a court of law doubt is cast on the testimony of a
witness. For example, the prosecution might show that the witness is a
known perjurer. This is a valid way of reducing the credibility of the
testimony given by the witness, and not Argumentum ad Hominem.
However, it doesn't demonstrate that the witness's testimony is false.

If you argue that someone should accept the truth of an assertion
because of that person's particular circumstances, then you have
committed the circumstantial form of argumentum ad hominem. For
example:

"It is perfectly acceptable to kill animals for food. How can you
argue otherwise when you're quite happy to wear leather shoes?"

This is an abusive charge of inconsistency, used as an excuse for
dismissing the opponent's argument. The fallacy can also be used as a
means of rejecting a particular conclusion. For example:

"Of course you would argue that positive discrimination is a bad
thing. You're white."

This particular form of Argumentum ad Hominem, when you allege that
someone is rationalizing a conclusion for selfish reasons, is also
known as "poisoning the well".

Argumentum ad ignorantiam

Argumentum ad ignorantiam means "argument from ignorance". The fallacy
occurs when it's argued that something must be true, simply because it
hasn't been proved false. Or, equivalently, when it is argued that
something must be false because it hasn't been proved true.

(Note that this isn't the same as assuming that something is false
until it has been proved true; that's a basic scientific principle.)

For example:

"Of course telepathy and other psychic phenomena do not exist.
Nobody has shown any proof that they are real."

Note that this fallacy doesn't apply in a court of law, where you're
generally assumed innocent until proven guilty.

Argumentum ad misericordiam

This is the Appeal to Pity, also known as Special Pleading. The
fallacy is committed when someone appeals to pity for the sake of
getting a conclusion accepted. For example:

"I did not murder my mother and father with an axe! Please don't
find me guilty; I'm suffering enough through being an orphan."

Argumentum ad populum

This is known as Appealing to the Gallery, or Appealing to the People.
You commit this fallacy if you attempt to win acceptance of an
assertion by appealing to a large group of people. This form of
fallacy is often characterized by emotive language. For example:

"If we allow religion in schools all our children will get brain-
washed."

Argumentum ad numerum

This fallacy is closely related to the argumentum ad populum. It
consists of asserting that the more people who support or believe a
proposition, the more likely it is that that proposition is correct.
For example:

"The vast majority of people in this country believe that capital
punishment has a noticable deterrent effect. To suggest that it
doesn't in the face of so much evidence is ridiculous.

"All I'm saying is that thousands of people believe in pyramid
power, so there must be something to it."

Argumentum ad verecundiam

The Appeal to Authority uses admiration of a famous person to try and
win support for an assertion. For example:

"Bultmann didn't believe in a physical resurrection of Christ"

This line of argument isn't always completely bogus; for example, it
may be relevant to refer to a widely-regarded authority in a
particular field, if you're discussing that subject. For example, we
can distinguish quite clearly between:

"Hawking has concluded that black holes give off radiation"

and

"Penrose has concluded that it is impossible to build an
intelligent computer"

Hawking is a physicist, and so we can reasonably expect his opinions
on black hole radiation to be informed. Penrose is a mathematician, so
it is questionable whether he is well-qualified to speak on the
subject of machine intelligence.

Argumentum ad antiquitatem

This is the fallacy of asserting that something is right or good
simply because it's old, or because "that's the way it's always been."
The opposite of Argumentum ad Novitatem.

"This interpretation has been accepted for hundreds of years.
It must be correct."

Argumentum ad novitatem

This is the opposite of the Argumentum ad Antiquitatem; it's the
fallacy of asserting that something is more correct simply because it
is new, or newer than something else.

Argumentum ad crumenam

The fallacy of believing that money is a criterion of correctness;
that those with more money are more likely to be right. The opposite
of Argumentum ad Lazarum.

Argumentum ad lazarum

The fallacy of assuming that someone poor is sounder or more virtuous
than someone who's wealthier. This fallacy is the opposite of the
Argumentum ad Crumenam.

Argumentum ad nauseam

This is the incorrect belief that an assertion is more likely to be
true, or is more likely to be accepted as true, the more often it is
heard. So an Argumentum ad Nauseam is one that employs constant
repetition in asserting something; saying the same thing over and over
again until you're sick of hearing it.

This is a common technique used by preachers (usually with a very shaky
argument!).

The fallacy of accident / Sweeping generalization / Dicto simpliciter

A sweeping generalization occurs when a general rule is applied to a
particular situation, but the features of that particular situation
mean the rule is inapplicable. It's the error made when you go from
the general to the specific. For example:

"Most aborigines have been in trouble with the law. You are an
aborigine so you must have been in trouble with the law as well."

This fallacy is often committed by people who try to decide moral and
legal questions by mechanically applying general rules.

Converse accident / Hasty generalization

This fallacy is the reverse of the Fallacy of Accident. It occurs when
you form a general rule by examining only a few specific cases which
aren't representative of all possible cases. For example:

"Jim Bakker was an insincere, immoral Christian. Therefore all
Christians are insincere."

Non causa pro causa

The fallacy of Non Causa Pro Causa occurs when something is identified
as the cause of an event, but it has not actually been shown to be the
cause. For example:

"I took an aspirin and prayed to God, and my headache disappeared.
So God cured me of the headache."

This is known as a false cause fallacy.

Post hoc ergo propter hoc

The fallacy of Post Hoc Ergo Propter Hoc occurs when something is
assumed to be the cause of an event merely because it happened before
that event.

This is another type of false cause fallacy.

Cum hoc ergo propter hoc

This fallacy is similar to post hoc ergo propter hoc. The fallacy is
to assert that because two events occur together, they must be
causally related. It's a fallacy because it ignores other factors that
may be the cause(s) of the events.

Petitio principii / Begging the question

This fallacy occurs when the premises are at least as questionable as
the conclusion reached. For example:

"Aliens are abducting innocent victims every week. The government
must know what is going on. Therefore the government is in league
with the aliens."

Circulus in demonstrando

This fallacy occurs if you assume as a premise the conclusion which
you wish to reach. Often, the proposition is rephrased so that the
fallacy appears to be a valid argument. For example:

This is the interrogative form of Begging the Question. One example is
the classic loaded question:

"Have you stopped beating your wife?"

The question presupposes a definite answer to another question which
has not even been asked. This trick is often used by lawyers in
cross-examination, when they ask questions like:

"Where did you hide the money you stole?"

Similarly, politicians often ask loaded questions such as:

"How long will this EU interference in our affairs be allowed to
continue?"

or

"Does the Prime Minister plan two more years of ruinous privatization?"

Another form of this fallacy is to ask for an explanation of something
which is untrue or not yet established.

Ignoratio elenchi / Irrelevant conclusion

The fallacy of Irrelevant Conclusion consists of claiming that an
argument supports a particular conclusion when it is actually
logically nothing to do with that conclusion.

For example, a Bhuddist may begin by saying that he will argue that
the teachings of the Bhuddahs are undoubtably true. If he then argues
at length that Bhuddism is of great help to many people, no matter
how well he argues he will not have shown that Bhuddist teachings are
true.

Sadly, such fallacious arguments are often successful because they
arouse emotions which cause others to view the supposed conclusion in
a more favourable light.

Equivocation / Fallacy of four terms

Equivocation occurs when a key word is used with two or more different
meanings in the same argument. For example:

"What could be more affordable than free software? But to make sure
that it remains free, that users can do what they like with it, we
must place a license on it to make sure that it will always be freely
redistributable."

One way to avoid this fallacy is to choose your terminology carefully
before beginning the argument, and avoid words like "free" which have
many meanings.

Amphiboly

Amphiboly occurs when the premises used in an argument are ambiguous
because of careless or ungrammatical phrasing.

Accent

Accent is another form of fallacy through shifting meaning. In this
case, the meaning is changed by altering which parts of a statement
are emphasized. For example, consider:

"We should not speak *ill* of our friends"

and

"We should not speak ill of our *friends*"

Be particularly wary of this fallacy in written communication, where
it's easy to mis-read the emphasis of what's written.

Fallacies of composition

One Fallacy of Composition is to conclude that a property shared by
the parts of something must apply to the whole. For example:

"The bicycle is made entirely of low mass components, and is
therefore very lightweight."

The other Fallacy of Composition is to conclude that a property of a
number of individual items is shared by a collection of those items.
For example:

"A car uses less petrol and causes less pollution than a bus.
Therefore cars are less environmentally damaging than buses."

Fallacy of division

The fallacy of division is the opposite of the Fallacy of Composition.
Like its opposite, it exists in two varieties. The first is to assume
that a property of some thing must apply to its parts. For example:

"You are studying at a rich college. Therefore you must be rich."

The other is to assume that a property of a collection of items is
shared by each item. For example:

"Ants can destroy a tree. Therefore this ant can destroy a tree."

The slippery slope argument

This argument states that should one event occur, so will other
harmful events. There is no proof made that the harmful events are
caused by the first event. For example:

"If we allow people to sing anything other than hymns in church
then we'll start having rock bands, then heavy-metal bands and
then the whole place will turn into disco."

"A is based on B" fallacies / "...is a type of..." fallacies / Fallacy of
the Undistributed Middle

These fallacies occur if you attempt to argue that things are in some
way similar, but you don't actually specify in what way they are
similar. Examples:

"Isn't history based upon faith? If so, then isn't the Bible also a
form of history?"

"Islam is based on faith, Christianity is based on faith, so isn't
Islam a form of Christianity?"

"Cats are a form of animal based on carbon chemistry, dogs are a
form of animal based on carbon chemistry, so aren't dogs a form of
cat?"

Affirmation of the consequent

This fallacy is an argument of the form "A implies B, B is true,
therefore A is true". To understand why it is a fallacy, examine the
truth table for implication given earlier. Here's an example:

"If I fall into the swimming pool, I get wet. I am wet, so I must
have fallen into the swimming pool."

This is the converse of Denial of the Antecedent.

Denial of the antecedent

This fallacy is an argument of the form "A implies B, A is false,
therefore B is false". The truth table for implication makes it clear
why this is a fallacy.

Note that this fallacy is different from Non Causa Pro Causa. That has
the form "A implies B, A is false, therefore B is false", where A does
not in fact imply B at all. Here, the problem isn't that the
implication is invalid; rather it's that the falseness of A doesn't
allow us to deduce anything about B.

"If I fall into the swimming pool, I get wet. I did not fall into
the swimming pool, therefore I am not wet."

This is the converse of the fallacy of Affirmation of the Consequent.

Converting a conditional

This fallacy is an argument of the form "If A then B, therefore if B
then A".

"If educational standards are lowered, the quality of argument seen
on the Internet worsens. So if we see the level of debate on the
net get worse over the next few years, we'll know that our
educational standards are still falling."

"If it's raining outside and I don't have an umbrella I get wet. So
if I get wet, then it's raining outside and I don't have an
umbrella."

This fallacy is similar to the Affirmation of the Consequent, but
phrased as a conditional statement.

Bifurcation

Also referred to as the "black and white" fallacy, bifurcation occurs
if you present a situation as having only two alternatives, where in
fact other alternatives exist or can exist.

Plurium interrogationum / Many questions

This fallacy occurs when someone demands a simple (or simplistic)
answer to a complex question.

Non sequitur

A non sequitur is an argument where the conclusion is drawn from
premises which aren't logically connected with it. For example:

"Since Egyptians did so much excavation to construct the pyramids,
they were well versed in paleontology."

Red herring

This fallacy is committed when someone introduces irrelevant material
to the issue being discussed, so that everyone else's attention is
diverted away from the points made, towards a different conclusion.

Reification / Hypostatization

Reification occurs when an abstract concept is treated as a concrete
thing.

Shifting the burden of proof

The burden of proof is always on the person asserting something.
Shifting the burden of proof, a special case of Argumentum ad
Ignorantiam, is the fallacy of putting the burden of proof on the
person who denies or questions the assertion. The source of the
fallacy is the assumption that something is true unless proven
otherwise.

"OK, so if you don't think the grey aliens have gained control of
the US government, can you prove it?"

It should be noted though, that if no "proof" can be offered for a
particular assertion, that in no way implies the assertion is not
true. There is a danger in any form of "positivism" which holds that
ONLY propositions that can be proved beyond doubt should be accepted
as true.

Straw man

The straw man fallacy is when you misrepresent someone else's position
so that it can be attacked more easily, then knock down that
misrepresented position, then conclude that the original position has
been demolished. It's a fallacy because it fails to deal with the
actual arguments that have been made.

"Textual criticism is wrong because it is inconsistent. Aleph and
B are accepted as the best manuscripts available but they disagree
so often in the gospels."

The above is straw man argument because the person that uses this
kind of argument does not really understand Textual Criticism. Textual
decisions are very complex and are not simply a matter of manuscript
preference. Aleph and B disgree in the gospels because B is Alexandrian
whereas Aleph is Western in the gospels and Alexandrian in the rest of
the NT.

The extended analogy

The fallacy of the Extended Analogy often occurs when some suggested
general rule is being argued over. The fallacy is to assume that
mentioning two different situations, in an argument about a general
rule, constitutes a claim that those situations are analogous to each
other.

This fallacy is best explained using a real example from a debate
about anti-cryptography legislation:

"I believe it is always wrong to oppose the law by breaking it."

"Such a position is odious: it implies that you would not have
supported Martin Luther King."

"Are you saying that cryptography legislation is as important as
the struggle for Black liberation? How dare you!"

Tu quoque

This is the famous "you too" fallacy. It occurs if you argue that an
action is acceptable because your opponent has performed it. For
instance:

"You're just being randomly abusive."

"So? You've been abusive too."

This is a personal attack, and is therefore a special case of
Argumentum ad Hominem.

Audiatur et altera pars

Often, people will argue from assumptions which they don't bother to
state. The principle of Audiatur et Altera Pars is that all of the
premises of an argument should be stated explicitly. It's not strictly
a fallacy to fail to state all of your assumptions; however, it's
often viewed with suspicion.

For example, many scholars reject the doctrine of inspiration and
inerrancy. This will of course dramatically affect their reasoning
and conclusions.

Ad hoc

There is a difference between argument and explanation. If we're
interested in establishing A, and B is offered as evidence, the
statement "A because B" is an argument. If we're trying to establish
the truth of B, then "A because B" is not an argument, it's an
explanation.

The Ad Hoc fallacy is to give an after-the-fact explanation which
doesn't apply to other situations. Often this ad hoc explanation will
be dressed up to look like an argument. For example, if we assume that
God treats all people equally, then the following is an ad hoc
explanation:

"I was healed from cancer."

"Praise the Lord, then. He is your healer."

"So, will He heal others who have cancer?"

"Er... The ways of God are mysterious."

Argumentum ad logicam

This is the "fallacy fallacy" of arguing that a proposition is false
because it has been presented as the conclusion of a fallacious
argument. Remember always that fallacious arguments can arrive at true
conclusions.

"Take the fraction 16/64. Now, cancelling a 6 on top and a six on
the bottom, we get that 16/64 = 1/4."

"Wait a second! You can't just cancel the six!"

"Oh, so you're telling us 16/64 is not equal to 1/4, are you?"

The "No True Scotsman..." fallacy

Suppose I assert that no Scotsman puts sugar on his porridge. You
counter this by pointing out that your friend Angus likes sugar with
his porridge. I then say "Ah, yes, but no true Scotsman puts sugar on
his porridge.

This is an example of an ad hoc change being used to shore up an
assertion, combined with an attempt to shift the meaning of the words
used original assertion. You might call it a combination of fallacies.